The Breuil-Mézard conjecture was first formulated for
2-dimensional representations in [1]}. Since then, it has
attracted a lot of attention. Kisin [2]} proved it when \(K ={\mathbb {Q}}_{p}\)  (and \(n = 2\) ) by making use of the \(p\) -adic local Langlands
correspondence for \(\operatorname{GL}_2({\mathbb {Q}}_{p})\)  and the (global) Taylor-Wiles-Kisin
patching argument. Sander [3]} and Paškūnas [4]} gave a
purely local alternative proof which has been extended later on
by Hu and Tan to nonscalar split residual representations [5]}.
For a general \(K\) , but still assuming \(n = 2\) , the conjecture
was proved by Gee and Kisin [6]} (see also
[7]}) when \(\lambda = (0, 0)\)  for each
embedding (which corresponds to potentially Barsotti–Tate
deformations).
